Theory and Applications of Vertical Variability Measures from Markov Chain Analysis

Finite Markov chain analysis has been used widely by sedimentologists in the search for fundamental patterns of lithological repetition that are statistically significant. The probability structure of a Markov model describes the relationship between adjacent events in a first-order process, but can be expanded to incorporate higher order memories. Simulations of stratigraphic successions from transition probabilities often are effective provided that any ancillary long-term trends also are accommodated. Markov stratigraphy can be used to produce multiple realizations of the internal structure of hydrocarbon reservoirs for use in fluid flow models. In addition, Markovian sequences have been modeled by synthetic seismograms. The discrimination of reflection frequency characteristics between synthetic seismograms from known facies types allows a lithostratigraphic classification of field seismic records. The Markovian statistics of vertical variability are applicable to selected problems of lateral prediction and simulation. The switch from the vertical to lateral direction is made possible by Walther's law, which states that lithologies that overlie one another must also have been deposited in adjacent tracts. Exceptions to Walther's law are caused by erosional breaks, but these are absorbed as a noise term within the probability model. Simulation of two- and three-dimensional models from Markovian vertical transitions must take into account the marked differences in scale and orientation that exist between the vertical and horizontal dimensions; however, some initial experiments indicate that results may be useful in applications ranging from pore network and rock fabric simulation to the modeling of local and regional geology. A finite Markov chain necessarily limits a simulation to a discretely stepped presentation of stratigraphic architecture; however, the discrete structure allows effective representations of bed boundaries and other sharp discontinuities. Geostatistical random functions can then be used to model internal variability of the Markovian events to refine the simulation.